In this paper, we present new sufficient conditions for the oscillation of first-order delay dynamic equations on time scales. We also present some examples to which none of the previous results in the literature can apply. | Turk J Math (2016) 40: 850 – 863 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article New oscillation tests and some refinements for first-order delay dynamic equations 1 2 ¨ ¨ Ba¸sak KARPUZ1,∗, Ozkan OCALAN ˙ Department of Mathematics, Faculty of Science, Tınaztepe Campus, Dokuz Eyl¨ ul University, Buca, Izmir, Turkey 2 Department of Mathematics, Faculty of Science, Akdeniz University, Antalya, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we present new sufficient conditions for the oscillation of first-order delay dynamic equations on time scales. We also present some examples to which none of the previous results in the literature can apply. Key words: Oscillation, delay dynamic equations, time scales 1. Introduction In this paper, we study the oscillation of the solution to the first-order delay dynamic equation x∆ (t) + p(t)x(τ (t)) = 0 for t ∈ [t0 , ∞)T , (1) where T is a time scale unbounded above with t0 ∈ T . We discuss (1) under the following assumptions. (C1) p ∈ Crd ([t0 , ∞)T , R+ ) . (C2) τ ∈ Crd ([t0 , ∞)T , T) is nondecreasing and satisfies the following conditions: (a) τ σ (t) ≤ t for all t ∈ [t0 , ∞)T . (b) limt→∞ τ (t) = ∞ . Before we proceed, let us recall some basic notions of the time scale concept. A time scale, which inherits the standard topology on R, is a nonempty closed subset of reals. Here, and later throughout this paper, a time scale will be denoted by the symbol T , and the intervals with a subscript T are used to denote the intersection of the usual interval with T. For t ∈ T , we define the forward jump operator σ : T → T by σ(t) := inf(t, ∞)T while the backward jump operator ρ : T → T is defined by ρ(t) := sup(−∞, t)T , and the graininess function µ : T → R+ 0 is defined to be µ(t) := σ(t) − t. A point t ∈ T is called right-dense if σ(t) = t and/or equivalently µ(t) = 0
